Question

A tridiagonal matrix is a square matrix with zeroes everywhere except on the main diagonal and the diagonals just above and below the main diagonal.

Let \(T\) be a real anti-symmetric tridiagonal matrix with elements \(t_{12}=c_{1}, t_{23}=c_{2}, \ldots\), \(t_{n-1 n}=c_{n-1}\). If \(n\) is even, compute det \(T\).

Answer 1

For an anti-symmetric matrix, the determinant is equal to the square of the magnitude of any of its eigenvalues. Since all eigenvalues of an anti-symmetric matrix are imaginary, the determinant of an anti-symmetric matrix is always positive. The determinant of a tridiagonal anti-symmetric matrix can be calculated as the product of the diagonal elements, which are the squares of the eigenvalues. Therefore, if \(n\) is even, the determinant of \(T\) is equal to the product of the squares of the eigenvalues, and is positive.